ing to the Jordan decomposition of real Radon measure on a locally compact space in [3] and [9], ... call E a normed M-module simply. Received June 27, 1959.
ON THE HAHN-BANACH TYPE THEOREM AND THE JORDAN DECOMPOSITION OF MODULE LINEAR MAPPING OVER SOME OPERATOR ALGEBRAS BY MASAMICHI TAKESAKI Introduction. In [6], Nachbin has shown the real Hahn-Banach extension property of C#(ίl), the space of all real valued continuous functions over a compact stonean space ίl, and recently Hasumi has proved the generalization of this result to complex case in [4]. While C(O) is considered, not only as a Banach space, but as a commutative algebra, then the extension problem of a module linear mapping over C(Ω) comes into our consideration. The same problem has been treated by Nakai in [7] which was independently presented from ours. On the other hand, Takeda and Grothendieck have shown the Jordan decomposition of self -adjoint linear functional on an operator algebra corresponding to the Jordan decomposition of real Radon measure on a locally compact space in [3] and [9], respectively. In the present note we shall show the extension property of C(ί2) for module linear mappings over C(ίl) and the generalization of Takeda-Grothendieck's result for self-adjoint module linear mappings over C(ίl). 1. Let M be a C*-algebra, M* the conjugate space of M and Jf** the second conjugate space of M. If π is a ^representation of M on a Hubert space H, then π is uniquely extended to the metric homomorphism π from M ** onto the weak closure of π(M) which is continuous for σ(M**, M*)-topology and xy are x°y for x,y^M respectively. Furthermore, if we consider a Banach algebra 5 instead of a normed M-module, then the slight modification of the above arguments points out that the second conjugate space J5** of B becomes a Banach algebra in two different manners. But we shall omit the detail. Next, we consider a certain linear mapping from a M-module E into M. A linear mapping θ from E into a normed M-module F called a left (resp. right) M-linear mapping if θ(ax) = α#(a:)
(resp. θ(xa) = θ(x)a)
for every a e M and a? e £7. If 0 is two-sided M-linear, it is called M-linear simply. Combining this definition and Theorem 1, we have
HAHN-BANACH TYPE THEOREM AND JORDAN DECOMPOSITION
5
LEMMA 3. If θ is a bounded M-linear mapping from E into F, then the u bitranspose θ = V of θ is M-linear. Proof. From the proof of Lemma 2, the mapping &->χofr is σ(M, Λf*)- and σ(E, J57*)-continuous for x^E. Hence we have θ(x°b) ~ θ(x)°b for x^E and T 6 e M. Using the σ(E, jE *)-continuity of the mapping x~^x°b, we get θ(x°V) = θ(x)°b for all x^E and b = (a1/2θ(ex)aί/2, τ> = 2
αα 1/2 , ^> - < a1/2xa1/2, φ+ > 5; 0
and similarly
